CHAPTER 1 Derivation of Telegrapher's Equations and Field-To

CHAPTER 1 Derivation of telegrapher’s equations and fi eld-to-transmission line interaction C.A. Nucci1, F. Rachidi2 & M. Rubinstein3 1University of Bologna, Bologna, Italy. 2Swiss Federal Institute of Technology, Lausanne, Switzerland. 3Western University of Applied Sciences, Yverdon, Switzerland. Abstract In this chapter, we discuss the transmission line theory and its application to the problem of external electromagnetic fi eld coupling to transmission lines. After a short discussion on the underlying assumptions of the transmission line theory, we start with the derivation of fi eld-to-transmission line coupling equations for the case of a single wire line above a perfectly conducting ground. We also describe three seemingly different but completely equivalent approaches that have been proposed to describe the coupling of electromagnetic fi elds to transmission lines. The derived equations are extended to deal with the presence of losses and multiple conductors. The time-domain representation of fi eld-to-transmission line coupling equations, which allows a straightforward treatment of non-linear phenomena as well as the variation in the line topology, is also described. Finally, solution methods in frequency domain and time domain are presented. 1 Transmission line approximation The problem of an external electromagnetic fi eld coupling to an overhead line can be solved using a number of approaches. One such approach makes use of antenna theory, a general methodology based on Maxwell’s equations [1]. Differ- ent methods based on this approach generally use the thin-wire approximation, in which the wire’s cross section is assumed to be much smaller than the minimum signifi cant wavelength. When electrically long lines are involved, however, the antenna theory approach requires prohibitively long computational times and high computer resources. On the WIT Transactions on State of the Art in Science and Eng ineering, Vol 29, © 2008 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/978-1-84564-063-7/01 4 Electromagnetic Field Interaction with Transmission Lines other hand, the less resource hungry quasi-static approximation [1], in which propagation is neglected and coupling is described by means of lumped elements, can be adopted only when the overall dimensions of the circuit are smaller than the minimum signifi cant wavelength of the electromagnetic fi eld. For many practical cases, however, this condition is not satisfi ed. As an example, let us consider the case of power lines illuminated by a lightning electromagnetic pulse (LEMP). Power net- works extend, in general, over distances of several kilometers, much larger than the minimum wavelengths associated with LEMP. Indeed, signifi cant portions of the frequency spectrum of LEMP extend to frequencies up to a few MHz and beyond, which corresponds to minimum wavelengths of about 100 m or less [2]. A third approach is known as transmission line theory. The main assumptions for this approach are as follows: 1. Propagation occurs along the line axis. 2. The sum of the line currents at any cross section of the line is zero. In other words, the ground – the reference conductor – is the return path for the currents in the n overhead conductors. 3. The response of the line to the coupled electromagnetic fi elds is quasi-transverse electromagnetic (quasi-TEM) or, in other words, the electromagnetic fi eld pro- duced by the electric charges and currents along the line is confi ned in the transverse plane and perpendicular to the line axis. If the cross-sectional dimensions of the line are electrically small, propagation can indeed be assumed to occur essentially along the line axis only and the fi rst assumption can be considered to be a good approximation. The second condition is satisfi ed if the ground plane exhibits infi nite conductiv- ity since, in that case, the currents and voltages can be obtained making use of the method of images, which guarantees currents of equal amplitude and opposite direction in the ground. The condition that the response of the line is quasi-TEM is satisfi ed only up to a threshold frequency above which higher-order modes begin to appear [1]. For some cases, such as infi nite parallel plates or coaxial lines, it is possible to derive an exact expression for the cutoff frequency below which only the TEM mode exists [3]. For other line structures (i.e. multiple conductors above a ground plane), the TEM mode response is generally satisfi ed as long as the line cross section is electrically small [3]. Under these conditions, the line can be represented by a distributed-parameter structure along its axis. For uniform transmission lines with electrically small cross-sectional dimensions (not exceeding about one-tenth of the minimum signifi cant wavelength of the exciting electromagnetic fi eld), a number of theoretical and experimental studies have shown a fairly good agreement between results obtained using the transmission line approximation and results obtained either by means of antenna theory or experiments [4]. A detailed discussion of the validity of the basic assumptions of the transmission line theory is beyond the scope of this chapter. However, it is worth noting that, by assuming that the sum of all the currents is equal to zero, we are WIT Transactions on State of the Art in Science and Eng ineering, Vol 29, © 2008 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Derivation of Telegrapher’s Equations 5 considering only ‘transmission line mode’ currents and neglecting the so-called ‘antenna-mode’ currents [1]. If we wish to compute the load responses of the line, this assumption is adequate, because the antenna mode current response is small near the ends of the line. Along the line, however, and even for electrically small line cross sections, the presence of antenna-mode currents implies that the sum of the currents at a cross section is not necessarily equal to zero [1, 3]. However, the quasi-symmetry due to the presence of the ground plane results in a very small contribution of antenna mode currents and, consequently, the predominant mode on the line will be transmission line [1]. 2 Single-wire line above a perfectly conducting ground We will consider fi rst the case of a lossless single-wire line above a perfectly conducting ground. This simple case will allow us to introduce various coupling models and to discuss a number of concepts essential to the understanding of the electromagnetic fi eld coupling phenomenon. Later in this chapter (Sections 4 and 5), we will cover the cases of lossy and multiconductor lines. The transmission line is defi ned by its geometrical parameters (wire radius a and height above ground h) and its terminations ZA and ZB, as illustrated in Fig. 1, where the line is illuminated by an external electromagnetic fi eld. The problem of interest is the calculation of the induced voltages and currents along the line and at the terminations. e e The external exciting electric and magnetic fi elds E and B are defi ned as the i i r r sum of the incident fi elds, E and B , and the ground-refl ected fi elds, E and B , determined in absence of the line conductor. The total fi elds E and B at a given point in space are given by the sum of the excitation fi elds and the scattered fi elds from the line, the latter being denoted as Es and Bs. The scattered fi elds are created by the currents and charges fl owing in the line conductor and by the corresponding currents and charges induced in the ground. Three seemingly different but completely equivalent approaches have been proposed to describe the coupling of electromagnetic fi elds to transmission lines. In what z E e e B 2a y C h ZA ZB x 0Lx x + dx Figure 1: Geometry of the problem. WIT Transactions on State of the Art in Science and Eng ineering, Vol 29, © 2008 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 6 Electromagnetic Field Interaction with Transmission Lines follows, we will present each one of them in turn. We will fi rst derive the fi eld-to- transmission line coupling equations (which are sometimes referred to as generalized telegrapher’s equations) following the development of Taylor et al. [5]. 2.1 Taylor, Satterwhite and Harrison model 2.1.1 Derivation of the fi rst fi eld-to-transmission line coupling (generalized telegrapher’s) equation Consider the single conductor transmission line of height h in Fig. 1. Applying Stokes’ theorem to Maxwell’s equation ∇E = –jwB for the area enclosed by the closed contour C yields ⋅=− ⋅ ∫∫∫Eldd jw BeSy (1) C S Since the contour has a differential width ∆x, eqn (1) can be written as hxx+∆ +∆ − + − ∫∫[Exzz ( xzExzz ,) (,)]d [ ExhEx xx (,) (,0)]d x 0 x h xx+∆ =− jBxzxzw∫∫ y (,)dd (2) 0 x (The coordinate y will be implicitly assumed to be 0 and, for the sake of clar- ity, we will omit the y-dependence unless the explicit inclusion is important for the discussion.) Dividing by ∆x and taking the limit as ∆x approaches zero yields ∂ hh ∫∫ExzzExhEx(,)d+−=− (,) (,0) jw Bxzz (,)d (3) ∂x zxx y 00 Since the wire and the ground are assumed to be perfect conductors, the total tangential electric fi elds at their respective surfaces, Ex(x,h) and Ex(x,0), are zero. Defi ning also the total transverse voltage V(x) in the quasi-static sense (since h << λ) as h =− Vx()∫ Ez (,)d xz z (4) 0 Equation (3) becomes d()Vx hhh =−jBxzzjBxzzjBxzzwww∫∫∫(,)d =−es (,)d − (,)d (5) dx yyy 000 where we have decomposed the B-fi eld into the excitation and scattered components.

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